Question

How do you evaluate `int (x^2)/(x^3+27)dx`?

Answer 1

The answer is `=1/3ln(|x^3+27|)+C`

Explanation:

Perform this integral by substitution,

Let `u=x^3+27`

`=>`, `du=3x^2dx`

The integral is

`int(x^2dx)/(x^3+27)=1/3int(du)/(u)`

`=1/3lnu`

`=1/3ln(|x^3+27|)+C`

Answer 2

`ln(abs[x^3+27])/3+c`

Explanation:

Problem:
`intx^2/(x^3+27)dx`

Substitute `u=x^3+27->dx=1/(3x^2)du`
`=1/3int1/udu`

Now solving `int1/udu`

This is a standard integral
`=lnu`

Plug in solved integrals:
`1/3int1/udu`
=`ln(u)/3`

Undo substitution `u=x^3+27`
=`ln(x^3+27)/3`

Add a constant lastly and absolute `x^3+27` because you cannot `ln` a negative number.